If the sum of m term is equal to n and sum of its n terms is equal to m then prove that sum of(m+n) terms is equal to -(m+n)
Given, sum of m terms = n
⇒ Sm = n
⇒ m/2 {2a + (m-1)d} =n
⇒ 2am +m(m-1)d = 2n ....(1)
Similarly, Sn = m
⇒ 2an + n(n-1)d= 2m .... (2)
On subtracting (2) from (1) we get
⇒ Sm - Sn = 2a(m-n) + {m(m-1) - n(n-1)} d = 2n- 2m
⇒ 2a (m - n) + (m-n)(m+n-1)d = -2(m-n)
⇒ 2a + (m+n-1)d = -2 .... (3)
Now, Sm+n = m+n/2 {2a+(m+n-1)d}
⇒ Sm+n = m+n/2 × (-2) [using (3)]
⇒ Sm+n = - (m+n)
[Hence proved]